arXiv subjects

Ricardo Ruiz-Baier

Publications and source records attributed to Ricardo Ruiz-Baier.

At least 19 recordsLinked to original sources

arXiv2026

Fully mixed virtual element schemes for a new model of steady-state poroelastic stress-assisted diffusion in the brain

We propose a fully mixed virtual element method for the numerical approximation of the coupling between linear poroelasticity equations with strong symmetry of total poroelastic stress (using the Hellinger--Reissner principle) and stress-altered solute diffusion (where diffusive flux depends on the poroelastic stress and nonlinearly on the concentration gradient). Because of the nonlinear coupling, the function spaces associated with the nonlinear diffusion sub-problem are of Banach type. To handle this structure, the solvability of both the continuous and discrete problems is established through a decoupled fixed-point strategy. The linear poroelasticity component is analysed using the theory for perturbed saddle-point problems, whereas the nonlinear diffusion problem, relies on the classical Minty--Browder theorem for monotone global operators. The existence of solutions for the fully coupled system is rigorously proven via Schauder's fixed-point theorem. Additionally, we establish rigorous a priori error estimates for the discrete scheme, successfully handling the strongly cross-coupled nonlinearities. These findings are supported by computational evidence, demonstrating that the formulation asymptotically recovers optimal convergence rates in practice. As a key contribution, both the numerical scheme and its underlying analysis prove to be robust with respect to the poromechanical parameters. Finally, several numerical examples are presented to illustrate the properties and applicability of the proposed scheme in the study of solute transport in the context of brain multiphysics.

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arXiv2026

A virtual element method for Kelvin--Voigt viscoelasticity

Considering the computational advantages of virtual element methods (VEM), this work employs a conforming VEM for the numerical approximation of the Kelvin--Voigt model of viscoelasticity. For clarity and simplicity, we focus on the primal formulation. The spatial discretization is carried out using the virtual element method, while the temporal discretization is handled via the {second-order Crank--Nicolson} scheme. We establish the well-posedness of both the semi-discrete and fully discrete problems and derive {\it a priori} error estimates. Several representative numerical examples are presented to validate the theoretical results and to demonstrate the effectiveness of the proposed formulation.

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arXiv2026

Unconditionally stable and energy conserving discretization of the dynamic von Kármán equations

A fully discrete approximation of the dynamic von Kármán equations combines nonconforming Morley finite element methods for spatial discretization with an energy conserving modified unconditionally stable Newmark second- order time-stepping scheme. Brouwer's fixed-point theorem establishes existence of a solution to the fully discrete scheme and further uniqueness and stability estimates follow for small loads. Optimal order a priori error estimates in the piecewise energy norm with quadratic convergence in time are derived for the fully discrete scheme. The results of the numerical experiments validate the theoretical error bounds.

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arXiv2026

Unified numerical analysis for thermoelastic diffusion and thermo-poroelasticity of thin plates

We investigate a coupled hyperbolic-parabolic system modeling thermoelastic diffusion (resp. thermo-poroelasticity) in plates, consisting of a fourth-order hyperbolic partial differential equation for plate deflection and two second-order parabolic partial differential equations for the first moments of temperature and chemical potential (resp. pore pressure). The unique solvability of the system is established via Galerkin approach, and the additional regularity of the solution is obtained under appropriately strengthened data. For numerical approximation, we employ the Newmark method for time discretization of the hyperbolic term and a continuous interior penalty scheme for the spatial discretization of displacement. For the parabolic equations that represent the first moments of temperature and chemical potential (resp. pore pressure), we use the Crank--Nicolson method for time discretization and conforming finite elements for spatial discretization. The convergence of the fully discrete scheme with quasi-optimal rates in space and time is established. The numerical experiments demonstrate the effectiveness of the 2D Kirchhoff--Love plate model in capturing thermoelastic diffusion and thermo-poroelastic behavior in specific materials. We illustrate that as plate thickness decreases, the two-dimensional simulations closely approximate the results of three-dimensional problem. Finally, the numerical experiments also validate the theoretical rates of convergence.

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arXiv2026

Stream function -- pressure virtual element methods for the Stokes--Darcy interface problem

This paper introduces a novel Virtual Element Method (VEM) for the coupled Stokes--Darcy system in primal-primal form. In the free-flow Stokes domain, we implement a stream function formulation that inherently satisfies the incompressibility constraint and reduces computational cost. Across the interface, mass conservation, normal stress balance, and the Beavers--Joseph--Saffman slip condition are enforced to couple the biharmonic stream function equation with the Darcy's pressure equation. Leveraging VEM's ability to handle general polygonal meshes, the proposed method naturally accommodates irregular interface geometries without requiring remeshing or adaptive refinement. The accuracy of the method is validated through several numerical simulations that include applications to dead-end filtration, and network flow in bioartificial organs.

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arXiv2026

Parameter-robust well-posedness and discretisation for coupled Darcy--Forchheimer and advection-diffusion-reaction equations

We adapt the recent theory for unique solvability of perturbed saddle-point problems in Banach spaces to the case of parameter-independent stability bounds. This also constitutes an extension to the Brezzi--Braess theory for parameter-robust stability of perturbed saddle-point problems from the Hilbert to the Banach setting. We apply the new abstract result to the mixed formulation of the advection-diffusion-reaction equation and tackle also its coupling with the Darcy--Forchheimer equations. The complete system is shown to be stable irrespective of the model parameters of permeability, Forchheimer coefficient, and reaction modulation. We discretise the problem with mixed finite element methods and show convergence in appropriately parameter-weighted norms. We also design operator-based preconditioners for the full system, utilising weighted norms that provide robustness with respect to model parameters.

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arXiv2026

Divergence-free unfitted finite element discretisations for the Darcy problem

We develop an unfitted compatible finite element discretisation for the Darcy problem based on $H(\mathrm{div})$-conforming flux spaces and discontinuous pressure spaces. The method is designed to preserve pointwise discrete mass conservation while remaining robust in the presence of arbitrarily small cut cells arising from unfitted meshes. Robustness is achieved by combining an $L^2$-stabilisation of the flux with an additional mixed-term stabilisation that enhances pressure control without destroying the local conservation structure. We consider both cell-wise (bulk) and face-based ghost-penalty realisations of the stabilisation. Mixed boundary conditions are handled by weak imposition of both flux and pressure traces on unfitted boundaries. We prove stability and a priori error estimates with constants independent of the cut configuration, and establish pressure-robust flux error bounds in the case of pure pressure boundary conditions. We also introduce an augmented Lagrangian variant that improves control of the conservation constraint and is amenable to efficient preconditioning strategies. Numerical experiments for a range of cut configurations, boundary-condition regimes and parameter choices confirm the theoretical results, demonstrating optimal convergence, cut-independent conditioning and mass conservation up to solver tolerance.

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arXiv2026

Linear poroelasticity with solid incompressibility: consistent formulation and scalable numerical solution

In this work we propose, by linearizing the equations of fully nonlinear poroelasticity, a consistent model in which only the solid phase is incompressible. This reformulation circumvents some inconsistency issues encountered in standard primal formulations of nonlinear poroelasticity while still retaining its key physical coupling mechanisms. We show a well-posed and consistent discretization strategy and also formulate scalable solvers based on a Schur complement formalism. A distinctive feature of the model is that it allows for a lowest order, inf-sup stable family of Finite Elements (FE) spaces. Numerical tests in two and three dimensions are provided to validate the proposed method and solver framework.

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arXiv2026

A Nitsche method for Navier--Stokes/generalized poroelasticity interface problems

We consider a time-dependent coupled Navier--Stokes/generalized poroelastic flow problem and propose a unified and monolithic finite element discretization based on implicit time stepping. To handle the fluid-structure interface we employ a Nitsche-type formulation. The resulting discrete problem is shown to be well-posed using the theory of differential-algebraic equations (DAEs) and the Banach fixed-point theorem. We prove stability and derive a priori error estimates for the fully discrete scheme. The stability and convergence of the method are ensured by a properly chosen penalty parameter independent of the mesh size. Numerical tests are presented to confirm the theoretical convergence rates and to illustrate the ability of the method to capture the coupled dynamics accurately.

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arXiv2026

A Residual Minimization approach for Nonlinear Partial Differential Equations set in Banach spaces

In this work, we propose and analyze a residual-minimization strategy for the numerical solution of nonlinear PDEs posed in Banach spaces. Given a finite-dimensional trial space and a suitably enriched discrete test space (of higher dimension than the trial space), we approximate the solution by minimizing the variational residual in a discrete dual norm. This minimization is equivalent to a nonlinear saddle-point formulation for the discrete solution in the trial space together with a residual representative in the test space. The latter provides a natural a posteriori error estimator, enabling automatic mesh adaptivity. To solve the resulting nonlinear saddle-point problem, we propose a Newton iteration whose linearized saddle-point system is symmetric, thereby guaranteeing solvability at each step. We take the $p$-Laplacian as a model problem and support the theoretical developments with representative numerical experiments, using standard $H^1$-conforming piecewise linear functions for the trial space, and lowest-order Crouzeix--Raviart functions for the test space.

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arXiv2026

Perturbed saddle-point problems in $\mathbf{L}^p$ with non-regular loads

In this work, we develop the discrete solvability analysis for perturbed saddle-point problems in Banach spaces with forcing terms regularised by means of a projector constructed using the adjoint of a weighted Clément quasi-interpolation. We take as driving example the linearised Poisson--Boltzmann (an advection-diffusion-reaction problem) in mixed form. We use perturbation arguments on the continuous and discrete levels and then derive a priori estimates that remain valid when the load that appears on the right-hand side of the "second" equation is in $\mathrm{H}^{-1}$. Further, we show a supercloseness result and {analyse convergence} of an adequate adaptation of Stenberg postprocessing for mixed advection equations with non-regular data. We provide numerical results that illustrate the convergence of the proposed scheme.

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arXiv2026

Four-field mixed finite elements for incompressible nonlinear elasticity

We present a stable finite element method for incompressible nonlinear elasticity based on a four-field mixed formulation involving the displacement, displacement gradient, first Piola--Kirchhoff stress and pressure. Unlike existing four-field mixed formulations, such as the compatible strain mixed finite element method (CSFEM), the proposed approach employs a discontinuous displacement field and requires no stabilisation in either 2D or 3D. A Newton--Raphson linearisation is derived and finite element pairs satisfying the relevant inf-sup conditions are identified. To recover accurate continuous displacement fields, an efficient postprocessing technique is further introduced. We establish the well-posedness of the linearised continuous problem together with a priori error estimates for the discrete formulation. Extensive numerical experiments in both 2D and 3D demonstrate optimal or even super convergence rates and enhanced robustness, particularly in 3D where CSFEM typically requires stabilisation.

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arXiv2026

A posteriori error analysis of a robust virtual element method for stress-assisted diffusion problems

We develop and analyse residual-based a posteriori error estimates for the virtual element discretisation of a nonlinear stress-assisted diffusion problem in two and three dimensions. The model problem involves a two-way coupling between elasticity and diffusion equations in perturbed saddle-point form. A robust global inf-sup condition and Helmholtz decomposition for $\mathbf{H}(\mathrm{div}, Ω)$ lead to a reliable and efficient error estimator based on appropriately weighted norms that ensure parameter robustness. The a posteriori error analysis uses quasi-interpolation operators for Stokes and edge virtual element spaces, and we include the proofs of such operators with estimates in 3D for completeness. Finally, we present numerical experiments in both 2D and 3D to demonstrate the optimal performance of the proposed error estimator.

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arXiv2026

Robust stability and preconditioning of Darcy-Forchheimer equations

We derive parameter-robust quasi-optimal error estimates for mixed finite element methods for the nonlinear Darcy--Forchheimer equations with mixed boundary conditions. Using the framework of operator preconditioning, we also design efficient block preconditioners for the linearised system, that exhibit robustness with respect to the coefficients that modulate permeability and inertia of the system. The properties of the formulation (parameter and mesh-size independence of the convergence rates) are illustrated by means of several numerical examples.

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arXiv2026

An ultra-weak three-field finite element formulation for the biharmonic and extended Fisher--Kolmogorov equations

This paper discusses a so-called ultra-weak three-field formulation of the biharmonic problem where the solution, its gradient, and an additional Lagrange multiplier are the three unknowns. We establish the well-posedness of the problem using the abstract theory for saddle-point problems, and develop a conforming finite element scheme based on Raviart--Thomas discretisations of the two auxiliary variables. The well-posedness of the discrete formulation and the corresponding a priori error estimate are proved using a discrete inf-sup condition. We further extend the analysis to the time-dependent semilinear equation, namely extended Fisher--Kolmogorov equation. We present a few numerical examples to demonstrate the performance of our approach.

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arXiv2026

Convergence analysis and adaptive computation of a Banach-space mixed finite element method for generalized bioconvective flows

We develop and analyse an adaptive fully mixed finite element method for stationary generalized bioconvective flows, where the Navier--Stokes equations with concentration-dependent viscosity are coupled with a conservation law for swimming microorganisms. The formulation introduces auxiliary variables including the trace-free velocity gradient, a symmetric pseudo-stress tensor, the concentration gradient, and a semi-advective microorganism flux, which also allows for a consistent treatment of Robin-type boundary condition. The variational problem is posed within a Banach space framework and reformulated as a fixed-point operator. Existence of solutions follows from Schauder's theorem, while uniqueness is obtained under suitable data assumptions. The discrete problem is constructed using Raviart--Thomas finite element spaces together with piecewise polynomial approximations on macroelement-structured meshes, and existence of discrete solutions is established via Brouwer's theorem. An a priori error analysis yields optimal convergence rates. We further derive a residual-based a posteriori error estimator and prove its reliability using global inf-sup conditions, Helmholtz decompositions, and suitable projection operators, while efficiency is ensured through localization techniques and bubble functions. Numerical experiments in two and three dimensions confirm the theoretical results, demonstrate the effectiveness of adaptive refinement for singular solutions and complex geometries with inclusions, and illustrate the robustness of the method for a bioconvective benchmark with plume formation governed by an Einstein--Batchelor-type viscosity law.

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arXiv2025

Analysis and virtual element discretisation of a Stokes/Biot--Kirchhoff bulk--surface model

We analyse a coupled 3D-2D model with a free fluid governed by Stokes flow in the bulk and a poroelastic plate described by the Biot-Kirchhoff equations on the surface. Assuming the form of a double perturbed saddle-point problem, the unique solvability of the continuous formulation is proved using Fredholm's theory for compact operators and the Babuska--Brezzi approach for saddle-point problems with penalty. We propose a stable virtual element method, establishing a discrete inf-sup condition under a small mesh assumption through a Fortin interpolant that requires only $H^1$-regularity for the Stokes problem. We show the well-posedness of the monolithic discrete formulation and introduce an equivalent fixed-point approach employed at the implementation level. The optimal convergence of the method in the energy norm is proved theoretically and is also confirmed numerically via computational experiments. We demonstrate an application of the model and the proposed scheme in the simulation of immune isolation using encapsulation with silicon nanopore membranes.

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arXiv2025

A velocity-vorticity-pressure formulation for the steady Navier--Stokes--Brinkman--Forchheimer problem

The flow of incompressible fluid in highly permeable porous media in vorticity - velocity - Bernoulli pressure form leads to a double saddle-point problem in the Navier--Stokes--Brinkman--Forchheimer equations. The paper establishes, for small sources, the existence of solutions on the continuous and discrete level of lowest-order piecewise divergence-free Crouzeix--Raviart finite elements. The vorticity employs a vector version of the pressure space with normal and tangential velocity jump penalisation terms. A simple Raviart--Thomas interpolant leads to pressure-robust a priori error estimates. An explicit residual-based a posteriori error estimate allows for efficient and reliable a posteriori error control. The efficiency for the Forchheimer nonlinearity requires a novel discrete inequality of independent interest. The implementation is based upon a light-weight forest-of-trees data structure handled by a highly parallel set of adaptive mesh refining algorithms. Numerical simulations reveal robustness of the a posteriori error estimates and improved convergence rates by adaptive mesh-refining.

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